= Hilbert functor
{c}
{title2=$\operatorname{Hilb}_{X/k}$}
For a <projective scheme> $X$ over a <field> $k$, the Hilbert functor assigns to a $k$-scheme $T$ the set of closed subschemes $W\subseteq X\times_kT$ which are of finite presentation, proper and flat over $T$. Pullback of families makes it a contravariant <functor>. After choosing a very ample <line bundle>, requiring a fixed fibre <Hilbert polynomial> $P$ gives the subfunctor represented by $\operatorname{Hilb}^P(X)$. The unrestricted functor is represented by the disjoint union of these <Hilbert schemes>.
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